Theorem iinexg 4612

Description: The existence of an indexed union. is normally a free-variable parameter in , which should be read (x). (Contributed by FL, 19-Sep-2011.)

Assertion

Ref	Expression
iinexg

Distinct variable group:   ,

Proof of Theorem iinexg

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiin2g 4363	. . 3
2	1	adantl 466	. 2
3		elisset 3120	. . . . . . . . 9
4	3	rgenw 2818	. . . . . . . 8
5		r19.2z 3918	. . . . . . . 8
6	4, 5	mpan2 671	. . . . . . 7
7		r19.35 3004	. . . . . . 7
8	6, 7	sylib 196	. . . . . 6
9	8	imp 429	. . . . 5
10		rexcom4 3129	. . . . 5
11	9, 10	sylib 196	. . . 4
12		abn0 3804	. . . 4
13	11, 12	sylibr 212	. . 3
14		intex 4608	. . 3
15	13, 14	sylib 196	. 2
16	2, 15	eqeltrd 2545	1

Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442  =/=wne 2652  A.wral 2807  E.wrex 2808  cvv 3109  c0 3784  |^|cint 4286  |^|_ciin 4331

This theorem is referenced by:  fclsval  20509  taylfval  22754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785  df-int 4287  df-iin 4333